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Theorem swrdccat1 11776
Description: Recover the left half of a concatenated word. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
swrdccat1  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. 0 ,  (
# `  S ) >. )  =  S )

Proof of Theorem swrdccat1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ccatcl 11745 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S concat  T )  e. Word  B )
2 lencl 11737 . . . . . 6  |-  ( S  e. Word  B  ->  ( # `
 S )  e. 
NN0 )
32adantr 453 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  NN0 )
4 nn0uz 10522 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4syl6eleq 2528 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
6 ccatlen 11746 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  =  ( ( # `  S
)  +  ( # `  T ) ) )
73nn0zd 10375 . . . . . . 7  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ZZ )
8 uzid 10502 . . . . . . 7  |-  ( (
# `  S )  e.  ZZ  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
97, 8syl 16 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) ) )
10 lencl 11737 . . . . . . 7  |-  ( T  e. Word  B  ->  ( # `
 T )  e. 
NN0 )
1110adantl 454 . . . . . 6  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  T
)  e.  NN0 )
12 uzaddcl 10535 . . . . . 6  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  ( # `  S ) )  /\  ( # `  T )  e.  NN0 )  ->  ( ( # `  S )  +  (
# `  T )
)  e.  ( ZZ>= `  ( # `  S ) ) )
139, 11, 12syl2anc 644 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( # `  S
)  +  ( # `  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
146, 13eqeltrd 2512 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  ( S concat  T ) )  e.  ( ZZ>= `  ( # `  S
) ) )
15 elfzuzb 11055 . . . 4  |-  ( (
# `  S )  e.  ( 0 ... ( # `
 ( S concat  T
) ) )  <->  ( ( # `
 S )  e.  ( ZZ>= `  0 )  /\  ( # `  ( S concat  T ) )  e.  ( ZZ>= `  ( # `  S
) ) ) )
165, 14, 15sylanbrc 647 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( # `  S
)  e.  ( 0 ... ( # `  ( S concat  T ) ) ) )
17 swrd0val 11770 . . 3  |-  ( ( ( S concat  T )  e. Word  B  /\  ( # `
 S )  e.  ( 0 ... ( # `
 ( S concat  T
) ) ) )  ->  ( ( S concat  T ) substr  <. 0 ,  ( # `  S
) >. )  =  ( ( S concat  T )  |`  ( 0..^ ( # `  S ) ) ) )
181, 16, 17syl2anc 644 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. 0 ,  (
# `  S ) >. )  =  ( ( S concat  T )  |`  ( 0..^ ( # `  S
) ) ) )
19 wrdf 11735 . . . . 5  |-  ( ( S concat  T )  e. Word  B  ->  ( S concat  T
) : ( 0..^ ( # `  ( S concat  T ) ) ) --> B )
20 ffn 5593 . . . . 5  |-  ( ( S concat  T ) : ( 0..^ ( # `  ( S concat  T ) ) ) --> B  -> 
( S concat  T )  Fn  ( 0..^ ( # `  ( S concat  T ) ) ) )
211, 19, 203syl 19 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( S concat  T )  Fn  ( 0..^ (
# `  ( S concat  T ) ) ) )
22 fzoss2 11165 . . . . 5  |-  ( (
# `  ( S concat  T ) )  e.  (
ZZ>= `  ( # `  S
) )  ->  (
0..^ ( # `  S
) )  C_  (
0..^ ( # `  ( S concat  T ) ) ) )
2314, 22syl 16 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( 0..^ ( # `  S ) )  C_  ( 0..^ ( # `  ( S concat  T ) ) ) )
24 fnssres 5560 . . . 4  |-  ( ( ( S concat  T )  Fn  ( 0..^ (
# `  ( S concat  T ) ) )  /\  ( 0..^ ( # `  S
) )  C_  (
0..^ ( # `  ( S concat  T ) ) ) )  ->  ( ( S concat  T )  |`  (
0..^ ( # `  S
) ) )  Fn  ( 0..^ ( # `  S ) ) )
2521, 23, 24syl2anc 644 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
)  |`  ( 0..^ (
# `  S )
) )  Fn  (
0..^ ( # `  S
) ) )
26 wrdf 11735 . . . . 5  |-  ( S  e. Word  B  ->  S : ( 0..^ (
# `  S )
) --> B )
2726adantr 453 . . . 4  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S : ( 0..^ ( # `  S
) ) --> B )
28 ffn 5593 . . . 4  |-  ( S : ( 0..^ (
# `  S )
) --> B  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2927, 28syl 16 . . 3  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  S  Fn  ( 0..^ ( # `  S
) ) )
30 fvres 5747 . . . . 5  |-  ( k  e.  ( 0..^ (
# `  S )
)  ->  ( (
( S concat  T )  |`  ( 0..^ ( # `  S ) ) ) `
 k )  =  ( ( S concat  T
) `  k )
)
3130adantl 454 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( ( S concat  T )  |`  (
0..^ ( # `  S
) ) ) `  k )  =  ( ( S concat  T ) `
 k ) )
32 ccatval1 11747 . . . . 5  |-  ( ( S  e. Word  B  /\  T  e. Word  B  /\  k  e.  ( 0..^ ( # `  S ) ) )  ->  ( ( S concat  T ) `  k
)  =  ( S `
 k ) )
33323expa 1154 . . . 4  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S concat  T
) `  k )  =  ( S `  k ) )
3431, 33eqtrd 2470 . . 3  |-  ( ( ( S  e. Word  B  /\  T  e. Word  B )  /\  k  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( ( S concat  T )  |`  (
0..^ ( # `  S
) ) ) `  k )  =  ( S `  k ) )
3525, 29, 34eqfnfvd 5832 . 2  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
)  |`  ( 0..^ (
# `  S )
) )  =  S )
3618, 35eqtrd 2470 1  |-  ( ( S  e. Word  B  /\  T  e. Word  B )  ->  ( ( S concat  T
) substr  <. 0 ,  (
# `  S ) >. )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   <.cop 3819    |` cres 4882    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   0cc0 8992    + caddc 8995   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045  ..^cfzo 11137   #chash 11620  Word cword 11719   concat cconcat 11720   substr csubstr 11722
This theorem is referenced by:  ccatopth  11778  ccats1swrdid  28213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-substr 11728
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